Constrained Optimization with a three variable function

Hi, I need help maximizing a function of three variables subject to a constraint. I understand how to do this for a function of two variables, but now I need to do it for a function of three variables.

If someone could point me in the right direction, recommend a webpage, etc., that would be great. Cursory googling has not helped.

Hi, I need help maximizing a function of three variables subject to a constraint. I understand how to do this for a function of two variables, but now I need to do it for a function of three variables.

If someone could point me in the right direction, recommend a webpage, etc., that would be great. Cursory googling has not helped.

Set your first order conditions = 0. Solve for to get it out of there. Then you solve the system of equations.

I forgot to add the zeros. Now I've edited them into my last post up there.

Now, I'm not sure what you mean by "solve the system of equations."

I'm guessing you mean I get them all into the same equation by substituting them in for 0?

In the example above, this leads me to:

Is that what you're talking about? And then I just solve for each variable to get its respective maximization formula?

So the maximization formula for c, for example, would be:

Is that right?

By the way thanks for that link. I had already come across it and it is one of the reasons why I have gotten as far as I have. But it does not deal with the three variable case so that is why I am asking here.

is a constant. Think of as the budget constraint. (sometimes denoted by m) is the purchasing power.

We want to maximize constrained by is your utility function, so the Lagrangian is a function that takes into account both the budget constraint and the utility function.

If we want to maximize a specific variable of the function we take the derivative w.r.t that variable and set it equal to Why do we do this? Because the derivative describes the slope of our function at a point. And when the slope we are at the zenith, maximum of the function.

Now, we have

Now sub into your other two equations and simplify them.

Now we have three equations and three unknowns.

Solve equation for say, then sub the obtained value of (a function of the other two variables) into equation Proceed in this manner to find the solutions, to your system of equations.

And then solving for any of these variables gives me the maximization formula for that variable? I.e. ? Is that it?

Oh, one last question. What if the constraint only involves two of the three variables. I.e. instead of it were only . In that case one of the first order condition equations would be without a lambda. Would that pose a problem for solving the system of equations? Thanks.

Oh, one last question. What if the constraint only involves two of the three variables. I.e. instead of it were only . In that case one of the first order condition equations would be without a lambda. Would that pose a problem for solving the system of equations? Thanks.

You will always have f.o.c. w.r.t. This is default.

In the two variable case you would have a f.o.c. w.r.t. x, and w.r.t. y because you are only trying to find the zenith for those two variables.